有限元模拟弹性位错的等效体力方法
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摘要
计算地震位错造成的位移场和应力场,对于估计大地震引起的后续地震活动发展趋势十分重要.本文提出在弹性位错问题的有限元模拟中,用等效体力代替位错源,从而在构建几何模型时不用包含断层,却可以处理包含任意复杂断层的问题,极大降低建模的难度.使用此方法,本文计算并讨论了在球形地球模型下2011年日本Tohoku-Oki特大地震对华北地区断层的影响,结果表明此次地震使华北主要断层趋于稳定.
Dislocation theory is well applied to calculate coseismic and postseismic effects.A key signature of the theory is that the solution of displacement is discontinuous.Various numerical methods can handle such discontinuous problems using a mesh which includes the discontinuous plane explicitly.However,generating such a mesh could be challenging and time consuming.We introduce an equivalent-bodyforce approach to handle discontinuities appearing in elastic dislocation theory.This approach gets rid of meshing the fault plane explicitly and simplifies the FEM modeling process.Based on Burridge and Knopoff′s work,we deduced a close-formed formula representing equivalent-bodyforce in FEM framework,and then compared our numerical results with Okada′s analytical solution in a test case in order to check the correctness of our formula and codes.At last,the 2011 Mw9.0Tohoku-Oki earthquake was studied.We compared our numerical results with GPS observations to check the correctness of our formula and codes again,and discussed the co-seismic effects in North China of this earthquake.In the test case,our numerical results differ from Okada′s analytical solution by less than3%in most computing regions.In modelling co-seismic effects of the 2011 Mw9.0Tohoku-Oki earthquake,our numerical results of displacement field agree well with GPS observations in bothdirection and magnitude.The co-seismic stress changes in North China are in east-west tension with a magnitude about 1kPa.The north-south compression is one order of magnitude smaller.The Coulomb failure stress changes on active faults in North China are negative which indicates more stable,except at the north end of the Tanlu fault zone where the Coulomb failure stress changes are about 100 Pa.Equivalent-bodyforce approach is applicable and accurate in FEM modeling.The 2011 Mw9.0Tohoku-Oki earthquake makes faults in North China more stable except the north end of the Tanlu fault zone.
Dislocation theory is well applied to calculate coseismic and postseismic effects.A key signature of the theory is that the solution of displacement is discontinuous.Various numerical methods can handle such discontinuous problems using a mesh which includes the discontinuous plane explicitly.However,generating such a mesh could be challenging and time consuming.We introduce an equivalent-bodyforce approach to handle discontinuities appearing in elastic dislocation theory.This approach gets rid of meshing the fault plane explicitly and simplifies the FEM modeling process.Based on Burridge and Knopoff′s work,we deduced a close-formed formula representing equivalent-bodyforce in FEM framework,and then compared our numerical results with Okada′s analytical solution in a test case in order to check the correctness of our formula and codes.At last,the 2011 Mw9.0Tohoku-Oki earthquake was studied.We compared our numerical results with GPS observations to check the correctness of our formula and codes again,and discussed the co-seismic effects in North China of this earthquake.In the test case,our numerical results differ from Okada′s analytical solution by less than3%in most computing regions.In modelling co-seismic effects of the 2011 Mw9.0Tohoku-Oki earthquake,our numerical results of displacement field agree well with GPS observations in bothdirection and magnitude.The co-seismic stress changes in North China are in east-west tension with a magnitude about 1kPa.The north-south compression is one order of magnitude smaller.The Coulomb failure stress changes on active faults in North China are negative which indicates more stable,except at the north end of the Tanlu fault zone where the Coulomb failure stress changes are about 100 Pa.Equivalent-bodyforce approach is applicable and accurate in FEM modeling.The 2011 Mw9.0Tohoku-Oki earthquake makes faults in North China more stable except the north end of the Tanlu fault zone.
引文
Arnold D N.1982.An interior penalty finite element method withdiscontinuous elements.SIAM Journal on Numerical Analysis,19(4):742-760.
Asano Y,Saito T,Ito Y,et al.2011.Spatial distribution and focalmechanisms of aftershocks of the 2011off the Pacific coast ofTohoku earthquake.Earth,Planets and Space,63(7):669-673.
Belytschko T,Black T.1999.Elastic crack growth in finiteelements with minimal remeshing.International Journal forNumerical Methods in Engineering,45(5):601-620.
Belytschko T,Moes N,Usui S,et al.2001.Arbitrary discontinuities infinite elements.International Journal for Numerical Methods inEngineering,50(4):993-1013.
Burridge R,Knopoff L.1964.Body force equivalents for seismicdislocations.Bulletin of the Seismological Society ofAmerica,54(6A):1875-1888.
Chen W T,Gan W J,Xiao G R,et al.2012.The impact of 2011Tohoku-oki earthquake in Japan on crustal deformation ofnortheastern region in China.Seismology and Geology(inChinese),34(3):425-439.
Daux C,Mo¨es N,Dolbow J,et al.2000.Arbitrary branched andintersecting cracks with the extended finite element method.International Journal for Numerical Methods in Engineering,48(12):1741-1760.
Dolbow J,Moes N,Belytschko T.2000.Discontinuous enrichmentin finite elements with a partition of unity method.FiniteElements in Analysis and Design,36(3-4):235-260.
Enescu B,Aoi S,Toda S,et al.2012.Stress perturbations andseismic response associated with the 2011 M9.0 Tohoku-okiearthquake in and around the Tokai seismic gap,central Japan.Geophysical Research Letters,39(13),doi:10.1029/2012GL051839.
Gonzalez-Huizar H,Velasco A A,Peng Z G,et al.2012.Remotetriggered seismicity caused by the 2011,M9.0 Tohoku-Oki,Japan earthquake.Geophysical Research Letters,39(10),doi:10.1029/2012GL051015.
Gracie R,Ventura G,Belytschko T.2007.A new fast finite elementmethod for dislocations based on interior discontinuities.InternationalJournal for Numerical Methods in Engineering,69(2):423-441.
Hiratsuka S,Sato T.2011.Alteration of stress field brought aboutby the occurrence of the 2011off the Pacific coast of Tohokuearthquake(Mw9.0).Earth,Planets and Space,63(7):681-685.
Oden J T,Kikuchi N,Song Y J.1982.Penalty-finite elementmethods for the analysis of Stokesian flows.Computer Methodsin Applied Mechanics and Engineering,31(3):297-329.
Okada Y.1992.Internal deformation due to shear and tensile faultsin a half-space.Bulletin of the Seismological Society ofAmerica,82(2):1018-1040.
Ozawa S,Nishimura T,Suito H,et al.2011.Coseismic andpostseismic slip of the 2011 magnitude-9 Tohoku-Oki earthquake.Nature,475(7356):373-376.
Shi Y L,Cao J L.2010.Some aspects in static stress changecalculation—case study on Wenchuan earthquake.ChineseJournal of Geophysics(in Chinese),53(1):102-110,doi:10.3969/j.issn.0001-5733.2010.01.011.
Simons M,Minson S E,Sladen A,et al.2011.The 2011magnitude9.0Tohoku-Oki earthquake:Mosaicking the megathrust fromseconds to centuries.Science,332(6036):1421-1425.
Sun W K,Okubo S,Fu G Y,et al.2009.General formulations ofglobal co-eismic deformations caused by an arbitrary dislocationin a spherically symmetric earth model-applicable to deformedearth surface and space-fixed point.Geophysical JournalInternational,177(3):817-833.
Sun W K,Zhou X.2012.Coseismic deflection change of the verticalcaused by the 2011Tohoku-Oki earthquake(Mw9.0).GeophysicalJournal International,189(2):937-955.
Takahashi H.2011.Static strain and stress changes in eastern Japandue to the 2011off the Pacific coast of Tohoku Earthquake,asderived from GPS data.Earth Planets and Space,63(7):741-744.
Wang M,Li Q,Wang F,et al.2011.Far-field coseismicdisplacements associated with the 2011Tohoku-oki earthquakein Japan observed by Global Positioning System.ChineseScience Bulletin,56(23):2419-2424.
Wang R J,Martín F L,Roth F.2003.Computation of deformationinduced by earthquakes in a multi-layered elastic crust—FORTRAN programs EDGRN/EDCMP.Computers&Geosciences,29(2):195-207.
Yukutake Y,Honda R,Harada M,et al.2011.Remotely-triggeredseismicity in the Hakone volcano following the 2011off thePacific coast of Tohoku Earthquake.Earth Planets and Space,63(7):737-740.
Zhao B,Wang W,Yang S M,et al.2012.Far field deformationanalysis after the Mw9.0 Tohoku earthquake constrained bycGPS data.Journal of Seismology,16(2):305-313.
Zhu G Z,Wang Q L.2005.Modeling of asymmetric earthquakedisplacement field of vertical Strike-slipe fault by using doublenode finite element technique.Journal of SeismologicalResearch(in Chinese),28(2):189-192.
陈为涛,甘卫军,肖根如等.2012.3·11日本大地震对中国东北部地区地壳形变态势的影响.地震地质,34(3):425-439.
石耀霖,曹建玲.2010.库仑应力计算及应用过程中若干问题的讨论———以汶川地震为例.地球物理学报,53(1):102-110,doi:10.3969/j.issn.0001-5733.2010.01.011.
朱桂芝,王庆良.2005.双节点有限元模拟直立走滑断裂地震位移场.地震研究,28(2):189-192.
Asano Y,Saito T,Ito Y,et al.2011.Spatial distribution and focalmechanisms of aftershocks of the 2011off the Pacific coast ofTohoku earthquake.Earth,Planets and Space,63(7):669-673.
Belytschko T,Black T.1999.Elastic crack growth in finiteelements with minimal remeshing.International Journal forNumerical Methods in Engineering,45(5):601-620.
Belytschko T,Moes N,Usui S,et al.2001.Arbitrary discontinuities infinite elements.International Journal for Numerical Methods inEngineering,50(4):993-1013.
Burridge R,Knopoff L.1964.Body force equivalents for seismicdislocations.Bulletin of the Seismological Society ofAmerica,54(6A):1875-1888.
Chen W T,Gan W J,Xiao G R,et al.2012.The impact of 2011Tohoku-oki earthquake in Japan on crustal deformation ofnortheastern region in China.Seismology and Geology(inChinese),34(3):425-439.
Daux C,Mo¨es N,Dolbow J,et al.2000.Arbitrary branched andintersecting cracks with the extended finite element method.International Journal for Numerical Methods in Engineering,48(12):1741-1760.
Dolbow J,Moes N,Belytschko T.2000.Discontinuous enrichmentin finite elements with a partition of unity method.FiniteElements in Analysis and Design,36(3-4):235-260.
Enescu B,Aoi S,Toda S,et al.2012.Stress perturbations andseismic response associated with the 2011 M9.0 Tohoku-okiearthquake in and around the Tokai seismic gap,central Japan.Geophysical Research Letters,39(13),doi:10.1029/2012GL051839.
Gonzalez-Huizar H,Velasco A A,Peng Z G,et al.2012.Remotetriggered seismicity caused by the 2011,M9.0 Tohoku-Oki,Japan earthquake.Geophysical Research Letters,39(10),doi:10.1029/2012GL051015.
Gracie R,Ventura G,Belytschko T.2007.A new fast finite elementmethod for dislocations based on interior discontinuities.InternationalJournal for Numerical Methods in Engineering,69(2):423-441.
Hiratsuka S,Sato T.2011.Alteration of stress field brought aboutby the occurrence of the 2011off the Pacific coast of Tohokuearthquake(Mw9.0).Earth,Planets and Space,63(7):681-685.
Oden J T,Kikuchi N,Song Y J.1982.Penalty-finite elementmethods for the analysis of Stokesian flows.Computer Methodsin Applied Mechanics and Engineering,31(3):297-329.
Okada Y.1992.Internal deformation due to shear and tensile faultsin a half-space.Bulletin of the Seismological Society ofAmerica,82(2):1018-1040.
Ozawa S,Nishimura T,Suito H,et al.2011.Coseismic andpostseismic slip of the 2011 magnitude-9 Tohoku-Oki earthquake.Nature,475(7356):373-376.
Shi Y L,Cao J L.2010.Some aspects in static stress changecalculation—case study on Wenchuan earthquake.ChineseJournal of Geophysics(in Chinese),53(1):102-110,doi:10.3969/j.issn.0001-5733.2010.01.011.
Simons M,Minson S E,Sladen A,et al.2011.The 2011magnitude9.0Tohoku-Oki earthquake:Mosaicking the megathrust fromseconds to centuries.Science,332(6036):1421-1425.
Sun W K,Okubo S,Fu G Y,et al.2009.General formulations ofglobal co-eismic deformations caused by an arbitrary dislocationin a spherically symmetric earth model-applicable to deformedearth surface and space-fixed point.Geophysical JournalInternational,177(3):817-833.
Sun W K,Zhou X.2012.Coseismic deflection change of the verticalcaused by the 2011Tohoku-Oki earthquake(Mw9.0).GeophysicalJournal International,189(2):937-955.
Takahashi H.2011.Static strain and stress changes in eastern Japandue to the 2011off the Pacific coast of Tohoku Earthquake,asderived from GPS data.Earth Planets and Space,63(7):741-744.
Wang M,Li Q,Wang F,et al.2011.Far-field coseismicdisplacements associated with the 2011Tohoku-oki earthquakein Japan observed by Global Positioning System.ChineseScience Bulletin,56(23):2419-2424.
Wang R J,Martín F L,Roth F.2003.Computation of deformationinduced by earthquakes in a multi-layered elastic crust—FORTRAN programs EDGRN/EDCMP.Computers&Geosciences,29(2):195-207.
Yukutake Y,Honda R,Harada M,et al.2011.Remotely-triggeredseismicity in the Hakone volcano following the 2011off thePacific coast of Tohoku Earthquake.Earth Planets and Space,63(7):737-740.
Zhao B,Wang W,Yang S M,et al.2012.Far field deformationanalysis after the Mw9.0 Tohoku earthquake constrained bycGPS data.Journal of Seismology,16(2):305-313.
Zhu G Z,Wang Q L.2005.Modeling of asymmetric earthquakedisplacement field of vertical Strike-slipe fault by using doublenode finite element technique.Journal of SeismologicalResearch(in Chinese),28(2):189-192.
陈为涛,甘卫军,肖根如等.2012.3·11日本大地震对中国东北部地区地壳形变态势的影响.地震地质,34(3):425-439.
石耀霖,曹建玲.2010.库仑应力计算及应用过程中若干问题的讨论———以汶川地震为例.地球物理学报,53(1):102-110,doi:10.3969/j.issn.0001-5733.2010.01.011.
朱桂芝,王庆良.2005.双节点有限元模拟直立走滑断裂地震位移场.地震研究,28(2):189-192.
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